Please help me with rental deposit

I’m a single mother currently living pay check to pay check. I’m paying rent weekly which makes it next to impossible to save up anything for the extra moving expenses such as a deposit( which in most cases is the same as rent). The rental market is very overpriced in New Orleans and especially in the surrounding areas. I’ve been patient with my living situation but it’s time for a change because I’m spending more money on rent by paying weekly. I will have no problem maintaining my rent and bills once i move it’s just hard getting my foot in the door. Please be a blessing to me and my son by helping us move before the new year. Thank you in advance